BKM's criterion for the 3D nematic liquid crystal flows via two velocity components and molecular orientations

TL;DR

This paper establishes a new criterion involving horizontal velocity gradients and molecular orientations that predicts the breakdown of solutions in 3D nematic liquid crystal flows, generalizing the Beale-Kato-Majda criterion.

Contribution

It introduces a novel blow-up criterion based on horizontal velocity and molecular orientation gradients, extending previous results to nematic liquid crystals and Navier-Stokes equations.

Findings

Provides a necessary and sufficient condition for solution breakdown.

Generalizes the classical BKM criterion to liquid crystal flows.

Offers new insights into the regularity criteria for complex fluid models.

Abstract

In this paper we provide a sufficient condition, in terms of the horizontal gradient of two horizontal velocity components and the gradient of liquid crystal molecular orientation field, for the breakdown of local in time strong solutions to the three-dimensional incompressible nematic liquid crystal flows. More precisely, let $T_{*}$ be the maximal existence time of the local strong solution $(u, d)$, then $T_{*}<+\infty$ if and only if \begin{align*} \int_{0}^{T_*} \big( \|\nabla_h u^h\|_{\dot{B}^0_{p,\frac{2p}{3}}}^q + \|\nabla d\|_{\dot{B}^0_{\infty,\infty}}^2 \big)dt = \infty\ \ \text{with}\ \ \ \frac{3}{p} +\frac{2}{q} = 2,\ \ \frac{3}{2} < p\leq\infty, \end{align*} where $u^{h}=(u^{1},u^{2})$, $\nabla_{h}=(\partial_{1}, \partial_{2})$. This result can be regarded as the generalization of the BKM's criterion in \cite{HW12}, and is even new for the three-dimensional incompressible Navier-Stokes equations.